BSTJ 59: 3. March 1980: Direct Distance Dialing: Call Completion and Customer Retrial Behavior. (Liu, K.S.)

Copyright © 1980 American Telephone and Telegraph Company
The Bell System Technical Journal
Vol. 59, No. 3, March 1980
Printed in U.SA.

Direct Distance Dialing: Call Completion and
Customer Retrial Behavior

By K. S. LIU

(Manuscript received September 14, 1979)

Most customers placing a direct- distance- dialing (ddd) call in the
United States are able to complete the call on the first attempt.
However, it is reasonable to expect that the probability of an initial
completion will be less than 1. When an initial attempt fails to
complete, a customer may decide to abandon his desired telephone
connection or to make one or more retrials. In general, a sequence of
one or more attempts may be initiated by a customer in an effort to
establish the desired connection. A study of ddd call completion and
retrials is important to provide an overall characterization of network
performance and customer behavior in setting up customers' desired
telephone connections. A survey adopting a two-stage stratified sam-
pling plan was undertaken to obtain ddd retrial statistics. Data
associated with sampled ddd calls that were originated from one of
890 switching entities in the Bell System network were collected for
a period of one week. The basic ddd retrial results reported here are
initial attempt disposition probabilities, retrial probabilities, number
of additional attempts, ultimate success probabilities, and distribu-
tion functions for retrial intervals following different types of incom-
plete initial attempts. Results of subclass analyses of retrial statistics
by originating and terminating classes of service (residence and
business) are also presented. Results obtained in this study are useful
in many network planning applications. An application of significant
importance is provision of a tool to evaluate the revenue and cost
impact of call completion improvement programs. A technique to
analyze the revenue and cost impact is outlined in the paper.

I. INTRODUCTION

Each day Bell System customers initiate about 780 million telephone
call attempts to be carried from one part of the telephone network to
the other. A very sophisticated network has been engineered to carry

295

both local and toll calls. Though only 7 percent of those calls are toll
calls, the number of toll calls is still very large indeed. For instance, on
an average business day in 1977 about 52 million toll call attempts
were placed by customers. Of those attempts, 36 million toll attempts
(roughly 70 percent) led to successful telephone connections, with the
other 16 million attempts resulting in failures because of the called
customer being busy or not in, circuit blockage, and so on. A customer
does not always succeed in establishing a desired connection on the
initial attempt. Some customers will abandon their effort to establish
desired connections after encountering incomplete attempts, others
will make additional attempts. Therefore, a desired connection may
lead to a sequence of more than one attempt, ultimately resulting in a
connection or abandonment by the calling customer. Whether each
attempt will result in a successful completion depends not only on
network-dependent variables such as engineered blocking level, but
also on customer-controlled variables such as telephone usage and on
interactions between the two groups of variables.

The main purpose of this study is to obtain results needed to
characterize both network performance and customer behavior not
only on initial attempts but also on possible subsequent attempts in
setting up a desired telephone connection. Retrial results are also
useful in many practical traffic engineering and network management
applications. As noted previously, roughly 70 percent of ddd call
attempts are completed. An important question is how customers react
after encountering the other 30 percent incomplete call attempts.
Equally important is the question: How often do customers eventually
establish their desired connections? It is shown in this paper that
answers to those and other questions can provide a useful tool in
evaluating the revenue and cost impact of efforts to improve network
call completions.

Each individual call attempt may be characterized by a complex
sequence of events involving actions and reactions by both customers
and the network. A detailed characterization of network performance
and customer behavior during the course of setting up a ddd call
attempt was given recently by Duffy and Mercer. 1 In that study, no
distinction was made on the attempt level, i.e., the order in a sequence
of attempts associated with the same desired connection. The results
therefore represent averages over all attempt levels. Also in that study,
very limited information was given about customer reattempts follow-
ing an unsuccessful attempt. Due to a restriction inherent in their data
sources, only information of up to three fast retrials occurring within
60 seconds of a previous observed incomplete attempt was obtained.
Thus, it was not possible to determine if any attempt had preceded
the first observed attempt or might follow the last observed attempt.

296 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1980

The only substantial amount of published retrial data in the Bell
System is what Wilkinson and Radnik 2 obtained in 1966 in a special
study of Automatic Message Accounting (ama) tapes associated with
toll calls originated at Norfolk, Virginia, and Levitttown, Pennsylvania.
In that study, data were obtained on whether and how many retrials
would follow an initial unsuccessful attempt. However, the reason for
failure of ddd attempts was not identified, so the description of the
retrial phenomena was incomplete. Without identifying the cause of
failure leading to subsequent retrials, it could not be determined under
what circumstances those retrial data might be used properly.

It is clear that a sufficient description will be obtained if one
monitors complete sequences of attempts from initial attempts to their
ultimate successes or abandonments. Two significant factors should
be considered in this type of approach. One is the amount of effort
needed in the design of proper instrumentation that can monitor many
customer lines simultaneously and gather detailed call set-up infor-
mation of every call originated from the lines being monitored. The
other factor is that, even with a proper instrumentation, a lengthy
survey would be required to obtain a scientific sample of data that
would characterize the ddd network. These two factors represented
major stumbling blocks; therefore, an alternative approach was sought.

To carry out the retrial study, a new retrial model is introduced in
which it is only necessary to determine the following: the disposition
of one attempt in a sequence of attempts, which number it is in the
sequence, how many attempts are made in the sequence, and the
ultimate success or failure of the attempt sequence. The parameters of
the model can be evaluated by combining service-observing data with
ama billing records from the same originating entities. In fact, for a
ddd attempt observed through the normal course of Dial Line Service
Observing (dlso), we can classify the disposition of an attempt. From
the called number recorded on the service observing card, the origi-
nating npa/nnx, and the time the call was made, each observed ddd
dlso attempt can be matched with a corresponding ama record. When
a match is obtained, a search on appropriate ama tapes will identify
all previous or subsequent attempts. Once complete sequences of
attempts are determined in this way, the attempt level — that is, initial
attempt, first retrial, etc. — for each observed dlso attempt in the
corresponding sequence of attempts can be determined.

The outline of this paper is as follows. Data collection and processing
procedure are explained in Section II. Section III presents the basic
initial attempt retrial results. Dependence of the retrial characteristics
upon both the originating and terminating classes of service is exam-
ined in Section IV. A class of service is used to designate a type of
telephone service provided to a group of customers. In this paper, two

DDD CALL COMPLETION 297

broad classes of service are considered, residence and business. Poten-
tial network revenue gains resulting from reducing ineffective attempts
are discussed in Section 5.1. Network revenue and cost impact of an
increase in the network completion ratio are analyzed in Section 5.2.
A special example of the impact of call waiting service is discussed in
Section 5.3. The results presented in the paper are summarized in
Section VI.

II. DATA COLLECTION AND DATA PROCESSING

At the outset of the sampling plan, it was recognized that existing
dlso bureaus should be considered as primary units and not be divided
further into smaller units to minimize possible administrative difficul-
ties. After comparing various possible sampling plans, it was discovered
that a primary stratification based upon the number of average annual
outgoing toll messages (aotm) per switching entity in a dlso bureau
would provide a substantial gain in precision for the network call
completion ratio (number of ddd messages/number of ddd attempts).
A two-stage random sampling plan with stratification was therefore
adopted. Under this sampling plan, all dlso bureaus were divided into
two strata. Within each stratum, the primary units, the dlso bureaus,
were selected with replacement and with probabilities proportional to
the sizes of the bureaus, as measured by the total number of aotm
from the observed switching entities in each dlso bureau. In the first
stage sample, five dlso bureaus were selected from the first stratum
and thirty bureaus from the second stratum. A smaller number of
bureaus was selected in the first stratum because of a smaller variance
for the network call completion ratio among the bureaus within that
stratum by design of the stratification. A total of 35 dlso bureaus in
17 Bell System operating companies were selected. The second stage
of sampling consisted of the selection of the actual observations of ddd
call attempts from each switching entity. Each observed dlso attempt
was selected with equal probability. However, the traffic carried by
one switching entity generally differed from that by others, therefore
each dlso observation from a switching entity had to be assigned a
traffic weight proportional to the annual outgoing toll traffic from that
entity for use in a proper estimation procedure. Estimates given in this
paper are accompanied by 90-percent confidence intervals if the sample
size is sufficiently large that the distribution of the ratio estimates used
may be assumed to be approximately normal. Confidence intervals are
not given when the sample size is too small for the normality assump-
tion.

For the 890 switching entities in the 35 dlso bureaus, the dlso data,
the corresponding ama records, and information about the class of
service for the calling and called customers were collected for a one-
week period in 1976.

298 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1980

As explained in Section I, the collected dlso data and ama data
were to be combined to match each dlso observation with an ama
record. A total of 12,658 dlso observations were successfully matched.
In this paper, the time period for attempts to be considered as part of
the same sequence are restricted to the same calendar day of the dlso
observation. (Effects due to an extension of the retrial period to include
the next calendar day have been found to be small.)

III. BASIC RETRIAL RESULTS

Out of 12,658 matched observations, 10,672 observations were found
to be initial attempts. The results presented in this paper are based
upon those attempts and the subsequent reattempts, if any.

Each of those 10,672 matched first attempts defines a desired con-
nection. For each desired connection, a sequence of attempts may be
initiated by the calling customer. The number of attempts, completion
probability, and retrial probability are listed in Table I for each
attempt level in those 10,672 sequences of attempts. It is seen that the
completion probability decreases steadily, while the retrial probability
increases steadily as the attempt level increases. The steady decrease
of the completion probability is a filtering effect. That is, as one
observes higher attempt levels, the sequences still "active" are those
directed to customers difficult to reach (or completion would have
occurred earlier) by customers who are more determined to get through
(or they would have abandoned earlier), so completion probability
decreases and retrial probability increases.

The probability of ultimate success in completing a desired connec-
tion is found to be 0.885 ± 0.009. This means 88.5 percent of the desired
connections are eventually established. Of the 88.5 percent successful
sequences, 75.5 percent of them succeed on the initial attempt, while

Table I — Completion and retrial probabilities by attempt

levels

Attempt Number of Completion Retrial

Level Attempts Probability Probability

1 10672 0.755 ±0.015 0.665 ± 0.042

2 1749* 0.510 ±0.051 0.743 ± 0.067

3 636* 0.415 ±0.061 0.793 ± 0.081

4 295* 0.377 ±0.052 0.871 ± 0.069
>5 386* 0.119 0.957

Average retrial probability = 0.719 ± 0.020
Average attempts/initial attempt = 1.29 ± 0.04
Ultimate success probability = 0.885 ± 0.009

* The number of attempts shown here is an unweighted count of the
number of remaining active sequences in the original 10,672 attempt se-
quences. To derive estimates such as the ratio of numbers of attempts on
subsequent attempt levels, each attempt must be weighted by an appropri-
ate traffic weight, as explained in the text.

DDD CALL COMPLETION 299

the other 13 percent succeed after an average of 1.79 additional
attempts. The call attempt time (excluding dialing time) is 18.2 seconds
for a completed attempt, and 32.1 seconds for an incomplete attempt.
The average call attempt time per successful sequence is (18.2 + 0.13/
0.885 X 1.79 X 32.1) = 26.6 seconds. The other 11.5 percent of the
sequences are abandoned after an average of an additional 0.52 at-
tempts. The average call attempt time per incomplete sequence is 1.52
X 32.1 = 48.8 seconds. Those incomplete sequences contribute a
substantial share of network load, or (0.115 X 48.8)/(0.115 X 48.8 +
0.885 X 26.6) = 19.3 percent of all nonconversation time in the ddd
network. They also represent a very significant loss of potential reve-
nue. (The revenue aspect is discussed further in Section V.) Overall,
there are 1.29 call attempts per initial attempt, and 1.45 call attempts
per message.

So far, the retrial results mentioned do not include the cause of
failure. Information given on dlso observations allows a detailed
classification of the call disposition of each observation. For the present
analysis, call dispositions are classified by the following five categories:
complete (comp), did not answer (da), busy (by), equipment blockage
and failure (eb&f), and everything else (Other). The eb&f category
encompasses all ineffective attempts due to network-caused problems
such as no circuit/reorder, no ring, and miscellaneous equipment
irregularities. The Other category is a mixture of several different
categories of calls classified as one of the following dispositions: did
not wait, correctly intercepted, no such number, customer-dialed
wrong number, and no response due to customer omitting an access
code.

Let q stand for one of the five dispositions listed above. The basic
quantities determined from the study upon which all subsequent
applications are based will be represented by the following:
p iq , the probability that the initial attempt disposition is q.
r q , the probability of a reattempt after an initial attempt with

disposition q.
L q , the average number of additional attempts after an initial
attempt with disposition q including sequences with no addi-
tional attempts.
S q , the probability that a connection is ultimately established given

that the initial attempt disposition is q.
The retrial quantities along with overall disposition probabilities p q
are summarized in Table II.
The following comments about the results are in order:
(i) The initial completion probability of 0.755 is substantially
higher than the overall completion probability of 0.69. For the
incomplete attempts, the change in disposition probabilities

300 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1980

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DDD CALL COMPLETION 301

goes in the opposite direction. For example, 19 percent of initial
attempts result in by or da, while 25.4 percent of all attempts
end up in these two dispositions.
(ii) In terms of retrial parameters, r g , L q , and S g , customer retrial
behavior is quite different after encountering different initial
dispositions. For example, the retrial probability ranges from
0.61 for calls encountering da conditions to 0.86 for calls
blocked due to network-caused problems.
(Hi) A surprisingly large 39 percent of the customers did not make
any reattempts at all after encountering initial das. As a result,
only 44 percent of sequences initially encountering a da were
eventually completed. Since 12.9 percent of the initial attempts
are das, this represents a 7.2-percent loss of all desired connec-
tions.
(iv) Customers tend to retry more often when encountering net-
work-related problems, though the chance of an ultimate suc-
cess is no better than that after by.
A very important retrial characteristic is how fast a customer makes
a reattempt. Retrial rate following eb&f is a critical factor influencing
how fast a network congestion may build up. An immediate retrial
following a by condition is likely to lead to another by. In the
remainder of this section, retrial time distribution is described in terms
of interarrival time between an initial attempt with a given incomplete
disposition q and the subsequent reattempt.

The mean and the median retrial times following initial by are 18.2
and 3.7 minutes, respectively; following da, 67.3 and 36.2 minutes,
following eb&f, 23.9 and 1.1 minutes. It is evident that in all three
cases the median retrial time is much smaller than the corresponding
mean retrial time. In terms of median retrial time, one can also see
three different time scales for retrials. For instance, the reattempt time
following an initial eb&f is relatively short (1.1 minutes), that following
an initial by is somewhat longer (3.7 minutes), and that following an
initial da is much longer (36.2 minutes).

If the interarrival time between an initial incomplete attempt and
the second attempt of the same desired sequence behaves like an
exponential distribution, then the ratio of the median retrial time to
the mean retrial time can easily be shown to be In 2 (=0.69). From the
values listed above, none of the three cases satisfies this criterion. In
fact, the three retrial time distribution functions following initial by,
da, and eb&f shown in Fig. 1 have been fitted to three functional
forms: (i) single exponential: (l-e" A '), (ii) two exponentials: p(l -
e~ xt ) + (1 -p)(l - e' 1 "), and (Hi) log-normal:

302 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1980

75 100 125 150 175

RETRIAL TIME INTERVAL IN MINUTES

Fig. 1 — Retrial time distribution functions.

225

250

The results of least-square fits are summarized in Table III. It can be
seen that the two-exponential form gives a better fit in all three cases.
In the case of by, the log-normal form also gives a reasonably small
mean-square deviation, considering the fact that it has only two
parameters instead of three for the two-exponential form.

IV. INFLUENCE OF CLASS OF SERVICE UPON RETRIAL
CHARACTERISTICS

The process of setting up a ddd call involves a complex interaction
among a calling customer, a called customer, and the telephone net-
work. The network, as complicated as it is, is outstanding in terms of
its performance to successfully carry call attempts. On the average,
only 2.2 percent of all ddd attempts fail to complete due to net-
work-caused problems. For the most part, whether a desired connec-
tion will be completed or not is determined by the characteristics of
the calling and called customers. The usage and capacity of a called
customer's telephone equipment and the degree of readiness to answer
an incoming call determine whether the mcoming call will be properly
answered, remain unanswered, or result in a busy condition. From the

DDD CALL COMPLETION 303

Table III— Results of least-square fits of retrial
distribution fun ctions following initial BY, DA, and EB&F

Initial Attempt Mean Square

Disposition Function Fitted Deviation

DA

Single exponential 1.0 X 10 ■

Log-normal 2.6 x 10 4

Two exponentials 1.5 x 10 *

Single exponential 5.5 x 10

Log-normal 1.9 x 10 3

Two exponentials 2.4 x 10 J

eb&f Single exponential 1.5 x 10

Log-normal 1.2 X 10

Two exponentials 4.9 X 10

viewpoint of the calling customer, his or her selection of time to place
the calls and his or her behavior during the call setup and any
subsequent reattempts certainly influence the outcome of these calls.

Customers in two important classes of service — residence and busi-
ness—tend to show differences in the above characteristics. These
customer characteristics may very well lead to substantial differences
in the basic retrial characteristics. In particular, one may reasonably
expect that a customer's class of service will influence retrial charac-
teristics associated with customer-related failures such as bys and das
that are beyond the control of the network. In this section, the retrial
parameters associated with calls originated from residence or business
customers to residence or business customers will be examined.

Estimates of the percentages of initial attempt dispositions of ddd
calls from residence to residence (R -> R), residence to business (R
-* B), business to residence (B -> R), and business to business (B ->
B) are summarized in Table IV. It can be seen from this table that
within the confidence intervals obtained in this study, the percentages
of initial attempts encountering by, eb&f, and "Other" are quite
similar in all four combinations of classes of service. The percentage of
calls resulting in initial completions or initial das varies drastically
from case to case. The initial completion percentages are 66.1 (R -*
R), 80.2 (R -* B), 62.7 (B -+ R), and 84.6 (B -► B). The corresponding
initial da percentages are 23.1, 6.4, 26.3, and 4.3. Clearly, the initial
completion percentage is higher for calls terminating at business lines
than those at residence lines. The main difference is due to a higher
percentage of das for calls directed to residence customers. The sums
of initial completion and da percentages are 89.2, 86.6, 89, and 88.9. In
all four combinations of classes of service, the initial da rate and the
initial completion rate practically complement each other.

Also listed in Table IV are the ultimate success probability, the
number of attempts per initial attempt, and the overall completion
ratio for calls between originating residence or business class of service
and terminating residence or business class of service. Calls terminat-
ing at residence lines, whether they are originated by residence or

304 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1980

Table IV — Initial attempt dispositions by originating and terminating

classes of service

Percentage

From Residence

From Business

Initial

Attempt

To

To

To

To

Disposition

Residence

Business

Residence

Business

Complete

66.1 ± 3.3

80.2 ± 3.5

62.7 ± 7.0

84.6 ± 3.0

Busy

5.6 ± 1.4

7.6 ± 2.2

6.0 ± 2.3

5.7 ± 1.6

Did not

answer

23.1 ± 2.9

6.4 ± 1.8

26.3 ± 6.9

4.3 ± 1.3

EB&F

2.2

1.4

1.6

2.5

Other

3.0

4.4

3.4

3.0

S

0.84 ± 0.02

0.92 ± 0.02

0.79 ± 0.06

0.94 ± 0.02

L

1.46 ± 0.14

1.29 ± 0.10

1.39 ± 0.19

1.18 ± 0.05

C

56.3 ± 3.1

73.2 ± 4.4

55.9 ± 4.2

80.0 ± 3.0

S = Ultimate success probability.

L = Number of attempts per initial attempt.

C = Overall completion ratio (number of messages/number of attempts).

business customers, have poor initial and overall completion ratios.
Almost 44 percent of all calls terminating at residence lines are
incomplete attempts. The percentage of desired connections that are
eventually established is 84 for R — ► R calls, and 79 for B — > R calls.
This means that 16 percent of desired connections from residence to
residence and 21 percent of desired connections from business to
residence are never completed.

In contrast, calls terminating at business lines have higher initial
and overall completion ratios and better ultimate success probabilities.
Though these parameters differ somewhat for residence-originated
calls and business-originated calls, the differences are much smaller
compared with the similar parameters for calls terminating at residence
lines. Overall, calls from business to business have the best chance of
completing, while those from business to residence have the worst.

Details of retrial characteristics for the four combinations of classes
of service are given in Table V. The initial retrial probability (r q ), the
number of additional attempts (L g ), and the ultimate success proba-
bility (S q ) have quite different values, depending upon the initial
attempt disposition. However, for a given initial disposition, there are
no drastic differences in most of those retrial characteristics among
calls placed between different combinations of the originating and
terminating classes of service. When a business customer encounters
an initial da, the probability of making one or more reattempts is 0.72
if he is calling another business customer. However if he is calling a
residence customer, he will retry only with a probability of 0.46.
Another interesting point is that residence customers tend to make a
slightly higher number of reattempts for calls intended to residence
customers when the initial attempt results in da.

The relative importance of bys and das in determining whether calls

DDD CALL COMPLETION 305

Table V — Retrial characteristics by originating and terminating
classes of service

Retrial
Charac-
teristic

A/ BY

r„ A

UD»

/"eb*f

L Klllh
iS KIHK

/"Other
L Other
Sother

Sample
size

From Residence

From Business

To
Residence

To
Business

To

Residence

0.71 ±0.11
1.50 ± 0.39
0.65 ± 0.10
0.58 ± 0.07
1.19 ± 0.19
0.44 ± 0.06
0.80
2.58
0.72
0.80
1.44
0.70
1562

0.74 ± 0.09
2.14 ± 0.89
0.67 ± 0.09
0.47 ± 0.12
0.94 ± 0.26
0.35 ±0.11
0.97
1.39
0.84
0.87
1.10
0.79
1252

0.75 ± 0.02
2.18 ± 1.25
0.65 ± 0.16
0.46 ± 0.14
0.79 ± 0.30
0.39 ± 0.16
0.86
1.71
0.67
0.51
0.64
0.27
750

To
Business

0.74 ± 0.14
1.63 ± 0.49
0.70 ± 0.14
0.72 ± 0.10
0.86 ± 0.13
0.51 ± 0.13
0.79
1.05
0.72
0.66
0.94
0.43
2826

r, = Retrial probability given that the initial attempt disposition is q.

L q = Average number of additional attempts given that the initial attempt disposition

is q. .....

S q = The ultimate success probability given that the initial attempt disposition is q.

will be completed is demonstrated in Table VI which lists the per-
centage of sequences that are abandoned eventually by customers
after encountering different initial incomplete attempts. For calls
terminating at residence lines, whether they are initiated by residence
or business customers, das are the main obstacles in preventing
completions of customers' desired connections. For calls terminating
at business lines, whether originated by residence or business cus-
tomers, das and bys are almost equally responsible for failures in
completing desired connections.

The most important implication of the above discussions is that, as
far as the initial attempt completion probability and the ultimate
success probability are concerned, the terminating class of service is
the dominant factor. The originating class of service has only a
secondary effect on the results.

V. APPLICATIONS

Retrial results obtained in this study are very important in providing
an overview of network performance and customer behavior in setting
up a desired telephone connection. They are also useful in many
practical applications. In traffic engineering, one has to take retrials
into account to properly relate a carried load to the offered load for a
given blocking objective. Especially in situations where repeated retri-
als may cause congestion in a local office or the network, information
about retrial characteristics is particularly needed. Similarly, knowl-
edge of customer behavior in reacting to network-caused failure should

306 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1980

Table VI — Percentages of sequences that are unsuccessful by
originating and terminating classes of service

Percentages of Sequences

That Are Unsuccessful

From Residence

From Business

Initial
Attempt

To

To

To

To

Disposition

Residence

Business

Residence

Business

Busy

2.0

2.5

2.1

1.7

Did not

12.9

4.2

16.0

2.1

answer

EB&F

0.6

0.2

0.5

0.7

Other

0.9

0.9

2.5

1.7

Total

16.4 ± 2.0

7.8 ± 2.0

21.1 ± 6.0

6.0 ± 2.0

be helpful in designing proper network management techniques to
minimize the impact of a temporary network problem.

Another application of significant importance is provision of a tool
to evaluate the revenue and cost impact of call completion improve-
ment programs. Several improvement programs have been adopted in
the Bell System operating companies to increase the network call
completion ratio, which is defined as the number of ddd messages
divided by that of ddd attempts. An initial analysis on the revenue
and cost impact of improving call completion on all ddd calls disre-
garding the originating and terminating classes of service is presented
in this section.

As noted in Section III, 11.5 percent of all desired connections are
never established. It will be shown that they represent a significant
loss of revenue. The amount of revenue loss varies from disposition to
disposition.

There are ways to recover some of the large revenue loss associated
with unsuccessful sequences of attempts. For instance, network-related
problems can be dealt with through a faster identification and correc-
tion of network problems. Customer-related problems (like by or da)
may be attacked by various marketing strategies like selling customers
new products or new services, or simply educating customers more
effectively. There undoubtedly will be an associated cost for each
program. It is not the purpose of the present paper to examine revenue
and cost effects for every possible strategy to find out the most cost-
effective way to improve the network completion ratio. Rather, the
following analyses are intended to demonstrate how the retrial results
reported here can be used to address two important aspects of the
problem, namely, network revenue and network cost of carrying ddd
messages and attempts. Questions like service charges and program
costs are outside the scope of the current investigation.

Revenue loss associated with each category of incomplete disposi-

DDD CALL COMPLETION 307

tions is evaluated in the first subsection. In the other two subsections,
two network improvement programs are discussed. These examples
help to demonstrate the usage of the previously presented retrial
quantities in other situations. One example discussed is the so-called
worth of completion question. It deals with the relationship of network
revenue gain to an increase in the network completion ratio. In the
other example, revenue effects of the call waiting service are discussed.
The call waiting service provides a subscriber with the option of
answering an incoming call even when he is already engaged in a
conversation.

5.1 Network revenue loss

Annual revenue loss for each category of ineffective initial attempts
can be computed as follows. The number of annual initial attempts
with disposition q is N X (Ni/iV) X p lg , where N is the number of
annual ddd attempts and Ni the number of annual initial ddd at-
tempts. The ratio Ni/N is determined from the data in Table I to be
0.775. Out of these sequences of attempts, Ni Xp iq X (1 - Sg) desired
connections are never established. Annual revenue loss for the category
is Ni X pig X (1 - S q ) X (revenue/message). The total annual ddd
revenue may be written as Ni ^pi q S q (revenue/message). The ratio
of annual revenue loss relative to this total revenue is 8.1 percent
associated with initial das, 2.2 percent for bys, and 0.7 percent for
eb&fs. The predominant revenue loss can be attributed to das and
bys. This loss, which results from customer-controlled failures, will be
recovered, not through direct improvements of the physical network,
but only through marketing efforts to reduce the incidence of das and
bys. It may be achieved by selling customers more lines or new vertical
services where available.

5.2 Worth of completion

To estimate the network revenue gain for a 1-percent increase in
the network completion ratio, it will be assumed that the increase is
made possible through the introduction of a call completion improve-
ment program. The program will result in an increase in completion
on the initial attempt for call sequences that would otherwise have
resulted in an initial disposition q. Let N\ q be the number of initial
incomplete attempts with disposition q that will be converted into
complete calls on the first attempts.* As a result of introducing the
network improvement program, those N\ q sequences will be affected.

* For dispositions other than da and by, the improvement program does not neces-
sarily convert all N iq initial attempts into completions, but into a mixture of completions,
das and bys. The following equations are for the DA and by dispositions. The modifi-
cations required for the other dispositions can easily be derived and are not shown here.

308 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1980

Before the conversion due to the program, N iq S q connections were
established, and an additional N\ q L q attempts were generated. After
the conversion, all N iq sequences are completed without any additional
attempts.

The total net additional gain consists of the net gain from the
additional N iq (1 — S q ) messages and the cost saving from Ni q (L q +
1 — S q ) incomplete attempts. The net gain per message is defined as
(revenue/message - cost/message). Therefore, the net worth (addi-
tional revenue — additional cost) may be written as

Net worth = iVi 9 (l — S q ) (net gain/message)

+ Ni q (L q + 1 — S q ) (cost/noncompletion)

= N\ q (l — S q ) (revenue/message)

— Ni q (l — S q ) (cost/message)

+ N\ q (L q + 1 - S q ) (cost/noncompletion).

If N is the total number of ddd attempts before the conversion and
c is the completion ratio, then Ni q is related to Ac, a change in the
completion ratio, by the following equation:

_ Nc + Ni q (l - S q ) _ fi lq (l -S q + cL q )

° N-N iq L q ° N - fi lq L g

or

NAc

Wi,-

1 - S q + (c + Ac)L 9 *

The net worth can now be rewritten as

Nbc

Net worth = — — -

1 - S q + (c + Ac)L 9

X [(1 — S q ) (revenue/message)

— (1 — S q ) (cost/message)

+ {L q + 1 — S q ) (cost/noncompletion)].

Thus, for a given change in the completion ratio, Ac, one can
compute both N iq and the net worth, where N iq is the number of initial
attempts with disposition q that need to be changed to achieve the
new completion ratio. The number of ineffective attempts with dis-
position q disregarding the attempt level, N q , that will be eliminated
in the process can also be estimated. N\ q /N q may be approximated by
the ratio of the number of initial dlso observations with disposition q
to that of the overall dlso observations with a disposition q at any

DDD CALL COMPLETION 309

attempt level, i.e., Nipi q /Np Q . An estimate of AT, is of practical interest,
since in general the result of an improvement program is measured in
terms of its effect on N q , not Ni q , and it is often desirable to know the
effect on revenues from such an improvement program.

Given appropriate revenue and cost figures per message and cost in
setting up an incomplete attempt, the cost and revenue effects for a
change in the completion ratio of ddd calls may be readily evaluated.

5.3 Call waiting service

The call waiting service is one of the vertical services available to
customers served by ess machines. It allows a subscriber to answer an
incoming call from a second calling party when he is already engaged
in a conversation. Normally, the second calling party would receive a
busy signal. With the call waiting service, the subscriber can respond
to the new incoming call, and thus change a would-be by attempt into
a complete call. If he decides to ignore the new incoming call, the
attempt would appear as a da instead of a by to the second calling
party. In the latter case, the second calling customer would presumably
hold on much longer than if he encountered a by signal. This choice
of response available to the subscriber introduces into the problem an
additional parameter /?, the fraction of time that the subscriber will
respond to the call-waiting signal. Let a be the marketing penetration
factor for the call waiting service. One can show in a similar analysis,
as given in the last two sections,*

Pi.by = (1 - a)pi,m

PlfiOW = Pl.COMP + /fopl.BY

]5l,DA = P1.DA + (1 - P)CtPW

. 1 + (api, m /p,)[fi(l - S DA ) - (Sby - Sda)]

c = c

1 + (acpi,BY/p 8 )((l - P)Lda - Z/by)

Incremental = NiapiBw[ p {1 _ Sda) _ ( Sby - Sda)]

messages
Incremental

ineffective = Niapi, B y[L B y + /?(!- Sby) - (1 - P)L D a]

and

attempts

Net worth = NiapiMlPO- ~ s ^) " (&Y - Sda)]
x (net gain/message)
+ [Lby + /?d - Sby)
— (1 — /?)L DA ] (cost/noncompletion)}.

* Here it is assumed that all incoming calls to call-waiting subscribers will not
encounter by conditions. In practice, a small percentage of incoming calls to call waiting
subscribers will still result in bys under certain conditions. The present analysis may be
generalized to include the effect of those remaining bys associated with calls to call
waiting subscribers.

310 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1980

The net worth is directly proportional to the market penetration
factor a. It is a linear function of the response factor ft. For a small ft,
the net worth may bcome negative. The reason is that, if subscribers
of the call waiting service do not respond to call-waiting signals, the
calling customers may perceive da conditions rather than would-be BY
conditions and will react accordingly. Since the ultimate success prob-
ability after da is lower than that after by, the trade-off may not be
favorable. Fortunately, if one uses reasonable estimates for various
parameters, the net worth will be positive if ft is greater than 0.25.
Initial results from a recent special study indicates that ft is approxi-
mately 0.63.

VI. SUMMARY

The important quantities needed to characterize retrial behavior are
the initial disposition probability p lq , the retrial probability r q , the
number of additional attempts L q , and the ultimate success probability
S q , where q represents any of the various possible dispositions. These
quantities, which are summarized in Table II, vary substantially for
different initial dispositions. Depending upon the intial attempt incom-
plete disposition, whether it be a by, da, or eb&f, the time interval
which a customer waits before initiating another attempt is quite
different.

Besides the traditional traffic engineering application of these retrial
parameters, several applications are included here to demonstrate the
usefulness of these retrial statistics. In addition, the technique used in
this paper has also been applied to several other problems in the
network planning area.

VII. ACKNOWLEDGMENTS

I would like to acknowledge the generous assistance of many Bell
System employees at Bell Laboratories, AT&T, and all participating
Bell System operating companies; without their individual efforts, this
study would have been an impossible task.

REFERENCES

1. F. P. Duffy and R. A. Mercer, "A Study of Network Performance and Customer

Behavior During Direct-Distance-Dialing Call Attempts in the U.S.A..," B.S.T.J.,
57, No. 1 (January 1978), pp. 1-33.

2. R. I. Wilkinson and R. C. Radnik, "The Character and Effect of Customer Retrials

in Intertoll Circuit Operations," 5th International Teletraffic Congress, New York,
1967.

DDD CALL COMPLETION 311